Subgroups — #26134693

Order 12032.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245124	C₂	1	2	yes	maximal, complemented
3	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244287	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#508	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	1	yes	characteristic
7	#15689161	C₂²	1	2	yes	—
8	#15689140	—	2	2	no	—
9	#15689097	—	2	2	no	—
10	#15689078	C₂²	1	1	yes	characteristic
11	#15688951	—	2	1	no	—
12	#15688949	—	2	1	no	—
13	#15688933	—	2	4	no	—
14	#15688898	—	2	1	no	—
15	C₁₆×Dic₄₇	—	2	1	no	—
16	C₂×#135×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#508	—	2	1	no	—
18	C₄₇×#508	—	2	1	no	—
19	C₄₇×#508	—	2	1	no	—
20	C₄₇×#508	—	2	1	no	—
21	C₂×Q₃₂×C₄₇	C₄	1	1	yes	characteristic, complemented
22	C₂×SD₃₂×C₄₇	C₄	1	2	yes	complemented
23	C₂×D₁₆×C₄₇	C₄	1	1	yes	characteristic, complemented
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625329	D₄	1	1	yes	characteristic
26	#13093983	—	2	2	no	—
27	#13093983	D₄	1	1	yes	characteristic
28	#13093964	—	2	1	no	—
29	#13093964	D₄	1	2	yes	—
30	C₂×C₄₇⋊C₁₆	—	2	1	no	—
31	#13093958	—	2	2	no	—
32	#13093956	—	2	2	no	—
33	#13093951	—	4	4	no	—
34	#13093939	—	4	4	no	—
35	#13093934	—	2	1	no	—
36	C₈×Dic₄₇	—	2	1	no	—
37	C₂×#41×C₄₇	—	2	2	no	—
38	#135×C₄₇	—	2	2	no	—
39	#135×C₄₇	—	2	2	no	—
40	#135×C₄₇	—	2	2	no	—
41	#135×C₄₇	—	2	2	no	—
42	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×SD₁₆×C₄₇	—	2	2	no	—
44	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	Q₃₂×C₄₇	—	2	1	no	—
47	Q₃₂×C₄₇	—	2	1	no	—
48	SD₃₂×C₄₇	—	2	2	no	—
49	SD₃₂×C₄₇	—	2	2	no	—
50	D₁₆×C₄₇	—	2	1	no	—
51	D₁₆×C₄₇	—	2	1	no	—
52	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
58	#41×C₄₇	—	4	2	no	—
59	#41×C₄₇	—	4	2	no	—
60	#41×C₄₇	—	4	2	no	—
61	#41×C₄₇	—	4	2	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₄₇	—	2	2	no	—
68	Q₁₆×C₄₇	—	2	2	no	—
69	SD₁₆×C₄₇	—	4	2	no	—
70	SD₁₆×C₄₇	—	4	2	no	—
71	D₈×C₄₇	—	2	2	no	—
72	D₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
78	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
79	C₂²⋊Dic₄₇	—	4	2	no	—
80	C₂²⋊Dic₄₇	D₈	1	2	yes	—
81	C₄⋊Dic₄₇	—	2	1	no	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄×Dic₄₇	—	2	1	no	—
84	C₄×Dic₄₇	—	4	2	no	—
85	#10625329	—	2	1	no	—
86	#10625329	D₈	1	2	yes	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄₇⋊C₁₆	—	2	2	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₄₇	—	4	2	no	—
96	Q₈×C₄₇	—	4	2	no	—
97	D₄×C₄₇	—	4	2	no	—
98	D₄×C₄₇	—	4	2	no	—
99	D₄×C₄₇	—	8	2	no	—
100	D₄×C₄₇	—	8	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	8	2	no	—
103	C₂×C₄×C₄₇	—	8	2	no	—
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	8	2	no	—
112	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
114	C₈×C₄₇	#121	1	1	yes	characteristic
115	C₈×C₄₇	#124	1	1	yes	characteristic
116	C₄₇⋊C₈	—	2	2	no	—
117	#13243	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	4	1	no	—
119	C₂²×C₄₇	—	8	2	no	—
120	C₂²×C₄₇	—	16	2	no	—
121	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
122	C₂²×C₄₇	#466	1	1	yes	characteristic
123	C₂²×C₄₇	#493	1	1	yes	characteristic
124	C₄×C₄₇	—	8	2	no	—
125	C₄×C₄₇	#390	1	1	yes	characteristic
126	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
127	C₄×C₄₇	#466	1	1	yes	characteristic
128	C₄×C₄₇	#495	1	1	yes	characteristic
129	Dic₄₇	—	4	2	no	—
130	Dic₄₇	—	16	2	no	—
131	C₂×#508	D₄₇	1	1	yes	characteristic
132	#2158	—	47	1	no	—
133	#2145	—	47	2	no	—
134	#2108	—	47	1	no	—
135	#1976	—	47	2	no	—
136	C₂×C₄₇	—	2	1	no	—
137	C₂×C₄₇	—	2	1	no	—
138	C₂×C₄₇	—	2	1	no	—
139	C₂×C₄₇	—	16	2	no	—
140	C₂×C₄₇	#1970	1	1	yes	characteristic
141	C₂×#135	C₂×D₄₇	1	2	yes	—
142	#508	—	2	1	no	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	C₂×Q₃₂	Dic₄₇	1	1	yes	characteristic, complemented
147	C₂×SD₃₂	Dic₄₇	1	2	yes	complemented
148	C₂×D₁₆	Dic₄₇	1	1	yes	characteristic, complemented
149	C₂×M₃₂	—	47	1	no	—
150	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
151	#427	—	47	2	no	—
152	#421	—	94	2	no	—
153	#410	—	94	2	no	—
154	#405	—	47	1	no	—
155	#367	—	94	1	no	—
156	#366	—	94	1	no	—
157	#360	—	94	4	no	—
158	#346	—	94	1	no	—
159	C₄×C₁₆	—	94	1	no	—
160	C₄₇	#13243	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	2	2	no	—
162	#135	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
169	C₂×M₁₆	—	47	1	no	—
170	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
171	#117	—	47	1	no	—
172	#117	—	94	2	no	—
173	Q₃₂	—	2	1	no	—
174	Q₃₂	—	2	1	no	—
175	SD₃₂	—	2	2	no	—
176	SD₃₂	—	2	2	no	—
177	D₁₆	—	2	1	no	—
178	D₁₆	—	2	1	no	—
179	M₃₂	—	47	2	no	—
180	M₃₂	—	94	1	no	—
181	C₂×C₁₆	—	94	1	no	—
182	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
188	#107	—	94	2	no	—
189	#106	—	94	2	no	—
190	#104	—	188	4	no	—
191	#99	—	188	4	no	—
192	#97	—	94	1	no	—
193	C₄×C₈	—	94	1	no	—
194	#41	—	4	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
203	Q₁₆	—	2	2	no	—
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	2	no	—
206	SD₁₆	—	4	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	2	2	no	—
209	M₁₆	—	47	2	no	—
210	M₁₆	—	94	1	no	—
211	C₂×C₈	—	94	1	no	—
212	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
213	C₂×C₈	#10625352	1	1	yes	characteristic
214	C₂×C₈	#10625352	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₄⋊C₄	—	94	1	no	—
219	C₄⋊C₄	—	188	2	no	—
220	C₂²⋊C₄	—	47	2	no	—
221	C₂²⋊C₄	—	188	2	no	—
222	C₄²	—	94	1	no	—
223	C₄²	—	188	2	no	—
224	C₁₆	—	94	2	no	—
225	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
226	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
227	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
228	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	D₈×D₄₇	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	376	2	no	—
242	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
243	C₂×C₄	#13093995	1	1	yes	characteristic
244	C₂×C₄	#13094020	1	1	yes	characteristic
245	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
246	C₂×C₄	#13094043	1	1	yes	characteristic
247	C₂×C₄	#13094043	1	1	yes	characteristic
248	C₈	—	94	2	no	—
249	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₈	D₄×Dic₄₇	1	1	yes	characteristic
251	C₈	#13093995	1	1	yes	characteristic
252	C₈	#13094011	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	D₈×Dic₄₇	1	1	yes	characteristic
257	C₂²	#15689356	1	1	yes	characteristic
258	C₂²	#15689498	1	1	yes	characteristic
259	C₄	—	8	2	no	—
260	C₄	—	188	2	no	—
261	C₄	—	752	2	no	—
262	C₄	#15689026	1	1	yes	characteristic
263	C₄	D₈×Dic₄₇	1	1	yes	characteristic
264	C₄	#15689356	1	1	yes	characteristic
265	C₄	#15689506	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#20244255	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689356
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting, prime radical	C₂×C₄₇×#508	C₂
socle	C₂×C₄₇	#1970
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#508	D₄₇
hypocenter	C₄₇	#13243
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20245172	C₂	semidirect
2. #20245124	C₂	semidirect
3. #20244984	C₂	semidirect
4. #20244287	C₂	semidirect
21. C₂×Q₃₂×C₄₇	C₄	semidirect
22. C₂×SD₃₂×C₄₇	C₄	semidirect
23. C₂×D₁₆×C₄₇	C₄	semidirect
146. C₂×Q₃₂	Dic₄₇	semidirect
147. C₂×SD₃₂	Dic₄₇	semidirect
148. C₂×D₁₆	Dic₄₇	semidirect
160. C₄₇	#13243	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245172	2
#20245124	2
#20244984	2
#20244287	2
C₂×C₄₇×#508	2
#13243	47